1102
DEANC. DOUGLASS AND DAVID W. MCCALL
Vol. 62
DIFFUSIOhTI N PARAFFIN HYDROCARBONS BY DEANC. DOUGLASS A N D DAVID W. MCCALL Bell Telephone Laboratories, Inc., M u r r a y Hill,New Jersey Receaued M a v IO, 1968
The Carr-Purcell spin-echo method has been used to measure the self-diffusion coefficients of the normal paraffins CsH1,, C6H14, C&6, CsH18, C&o, CloH22, C18H38 and C32He6. The measurements were made over a temperature range in order to obtain activation energies. Within experimental error the plots of log D vs. 1/T are linear. As expected the diffusion coefficients decrease with increasing molecular weight.
The activation energies increase with increasing molecular weight.
It .is proposed that the elementary diffusion rocess involves the translation of an extended molecule parallel t o its chain axis. A reduced temperature plot of the difPusion coefficient clusters the data in an interesting manner and the diffusion coefficients for all the hydrocarbons extrapolated t o their respective critical points are approximately equal.
I. Introduction A satisfactory molecular theory of transport properties for liquids has not yet been developed. This is due, in part, to the inherently complicated nature of the liquid state but it is also important that experimental investigations of transport properties, other than viscosity, have been carried out in too few systems. For example, Johnson and Babbl were able to report experimental values for selfdiffusion coefficients in only ten non-electrolytes and to our knowledge only two liquids2 have been added t o this list since their review. The present paper is a comparative study of self-diffusion in several paraffin hydrocarbons. The discovery of nuclear magnetic resonance spin-echoes by Hahn3 and the ensuing theoretical a i i a l y s i ~of~ ~the ~ experiment led t o a new technique for measuring self-diffusion coefficients which does not require the presence of unusual isotopes. Spin-echo measurements may be used t o measure self-diffusioncoefficients rapidly and under a variety of experimental conditioiis. This method is employed in the present study using the proton magnetic resonance. Probably the only theory of liquid diffusion that has established contact with the experimentalists is the Eyring theory of rate,processes.6-8 This is because the theory provides simple equations which have been quite successful in correlating experimental data and the parameters of the equ a t'lolls are interpreted in terms of a simple model for the diffusion process. According to Eyring's theo.ry8 D = X2((kll'/h) exp( - A F * / R ) (1) where AF* is the free energy of activation and X 2 is the mean squibre jrinip lengtth. This equstion will form the hasis for the discussion of thc d:hi reported herein. 11. Analysis of the Experiment The essentials of the spin-echo experiment have been clearly described by Hz~h11.~The basic ex-
periment for a "90-180'" pulse sequence may be qualitatively described as follows. The specimen being studied is contained in a coil whose axis is perpendicular to a n externally applied magnetic field. At equilibrium the nuclear magnetic moments tend to be aligned parallel with the external field, forming a macroscopic magnetization. A short, intense burst of radiofrequency energy a t the resonant frequency is applied t o the specimen through the coil and the magnetization is rotated 90' relative to H,. After the pulse the msgnetiaation vector undergoes free precession about the field direction a t a frequency proportional to the magnetic field strength. This precessing magnetization induces a voltage in the coil. However, because the field is never strictly uniform over the sample, the total magnetization vector of the sample must be thought of as the resultant of many partial magnetization vectors, each precessing at a frequency slightly different from the others. That is to say, since there is a distribution in field over the sample there is also a distribution in frequency. Immediately after the 90O-pulse a11 of the partial magnetization vectors are in phase. However, because of their different precession frequencies, they rapidly dephase and the voltage induced in the coil decreases to zero. At a later time, r , the component of each partial magnetization vector which is perpendicular to the coil axis is rotated 180" by a 18O0-pulse. This is equivaleiit to a reflection of the partial magnetization vectors in the plane determined by the coil axis and the field direction. If two partial magnetization vectors have their relative phase angle increasing before the 180"-pulse it will be decreasing after the 180O-pulse and vice versa. Provided we assume that the rate of precession of each partial magnetization vector remains constant overf the period 27, the 180O-pulse will bring all the partial magnetization vectors into phase a t time 2i(where r is the time between the 90"- and 180'pulses). During the time the magiiet,ization vectors are in phase a voltage, called a "spin-echo," is (1) P. A. Johnson and A. L. Babb, Chem. Reus., 66, 387 (1956). induced in the coil. (2) E. Fishman, THISJOURNAL, 59, 469 (1955). If the sample is in a non-uniform magnetic field (3) E. L. Hahn, Phys. Rev., 80,580 (1950). (4) H.Y.Carr and E. M. Purcell, ibid., 94, 630 (1954). and diffusion occurs, then the average magnetic (5) T. P. Das and A. K. Saha, ibid.. 93, 749 (1954). field seen by a given nucleus between times 0 and (6) S. Glasstone, K. J. Laidler and H. Eyring, "The Theory o r Rate Processes," MoGraw-HI11 Book C o . , Inc., New York. N. Y., r is not necessarily the same as the average magnetic field seen by this nucleus between times i1941. (7) J. 0. Hirsohfelder, D. Stevenson and H. Eyring, J . Chsm. and 27. Thus the phase gain between 0 and T is Phvs., 5, 896 (1937). not, in general, the same as the phase loss between (8) J. F. Kincaid, H. Eyring and A. E. Steam, Chem. Reus., 38, 301 r and 27. and the number of nuclei brought back (1941). into phase a t 27 is reduced. Thus the effect of (9) E. L. Hahn, Phy6zCS Today, 6, 4 (1953).
DIFFUSION IN PARAFFIN HYDROCARBONS
Sept., 1958
diffusion is t o reduce the voltage induced in the coil by the "spin-echo." By assuming a random walk model of diffusion and a uniform field gradient over the sample Carr and Purcel14 were able t o derive an expression equivalent to Vmax = A(27) exp( - 2 y 2 G 2 D r 3 / 3 )
(1)
where V,,, is the maximum amplitude of the echo voltage and A(27) is assumed to be independent of the field gradient G, and the diffusion coefficient D. Thus A(27) may be measured with G = 0 and D is then determined from the slope of a plot of ln(V,,,/A) vs. r3 when G # 0. The result of Carr and Purcell may be obtained without reference to a random walk model of diffusion in the following manner. The phase accumulated by a precessing magnetic moment a t time t 3 r where T is the time a t which the 180'pulse is introduced is given by
If the sample resides in a uniform magnetic field gradient
+ Gz(t)
H ( t ) = Ho
distance a precessing moment moves during the t,ime of the experiment (27), then c = (2?rtD)-'/2exp( -Y2/4Dt)
= Y
{G
K
@ = @O
-G
z ( t ) dt
+ Gy
l
z ( t ) dt
{K
z ( t ) dt
-
+ Ho(27 - t ) /
l
(3)
where p ( @ ) is the probability density of moments having relative phase @. p ( @ ) can be computed directly from equation 2 by first making the substitution Y ( t ) = x ( t ) - xo where xo is the initial position of the precessing moment being followed. Then
(4)
and xo are independent variables and therefore P(@)
=
d @ l )Nzo)
(5)
Y ( t ) is simply the position of a diffusing particle which starts a t Y = 0 a t time t = 0. Therefore, the probability density for Y ( t )is a solution of the usual diffusion equation
3C=,- P bt
C
bY2
If the size of the sample is large compared to the
(7)
s
(9)
av = 2Dt
Substitution of equation 9 into equation 8 and integration gives
